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Chapter 6: Q24MP (page 338)

鈭�r脳苍诲蟽over the entire surface of the hemispherex2+y2+z2=9,z鈮�0

where r=xi+yj+zk.

Short Answer

Expert verified

The solution is 鈭�鈭�T鈥�r脳苍诲蟽=54蟺.

Step by step solution

01

Given Information.

The given information is,

x2+y2+z2=9,z鈮�0

02

Definition of Divergence Theorem.

The divergence theorem, often known as Gauss' theorem or Ostrogradsky's theorem, is a theorem that connects the flow of a vector field across a closed surface to the field's divergence in the volume enclosed.

03

Find the solution.

Use the divergence theorem 鈭�T鈥�鈭�脳VdT=鈭�鈭�T鈥�V脳苍诲蟽, where 鈭�Tis the surface area that encloses the volume T.

鈭�脳r=鈭�rx鈭�x+鈭�ry鈭�y+鈭�rz鈭�z

role="math" localid="1657354820813" =1+1+1

=3

Here,鈭�蟽 is the entire surface of the hemisphere of radius3 centered at (0,0,0), soT is just the volume of the sphere.

鈭�鈭�T鈥�r脳苍诲蟽=鈭�鈭�T鈥�鈭�脳rdT

=3鈭�T鈥�dT

=31243(蟺)(3)3

=54蟺

Hence, the solution is 鈭�鈭�T鈥�r脳苍诲蟽=54蟺.

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Most popular questions from this chapter

Use either Stokes' theorem or the divergence theorem to evaluate each of the following integrals in the easiest possible way.

鈭�surface蟽鈥�curl鈦�(x2i+z2j鈭�y2k)鈭�苍诲蟽, where 蟽 is the part of the surface z=4鈭�x2鈭�y2above the (x,y) plane.

If the temperature is T=x2-xy+z2, find

(a) The direction of heat flow at (2,1, -1);

(b) The rate of change of temperature in the direction j-kat(2,1,-1)

Find the derivative of xy2+yz at (1,1,2) in the direction of the vector 2i-j+2k.

鈭�V脳苍诲蟽over the entire surface of the volume in the first octant bounded byx2+y2+z2=16and the coordinate planes, whereV=x+x2鈭�y2i+(2xyz鈭�2xy)j鈭�xz2k

Verify that the force field is conservative. Then find a scalar potential 蠁 such thatF=-鈭�蠁,F=yi+xj+k ,

K = constant.
See all solutions

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